Notes on a Family of Inhomogeneous Linear Ordinary Differential Equations

In the note, the authors find several simple and explicit forms for a family of inhomogeneous linear ordinary differential equations studied in “D. Lim, Differential equations for Daehee polynomials and their applications, J. Nonlinear Sci. Appl. 10 (2017), no. 4, 1303–1315; Available online at http://dx.doi.org/10.22436/jnsa.010.04.02”. 1. Motivation and main results In [3, Theorem 1], it was obtained inductively and recurrently that the differential equations ∂ F (t, x) ∂ tn = 1 tn [ n ∑ i=0 ( ai(n, x) + bi(n, x) ln(1 + t) )( t 1 + t )i] F (t, x), n ≥ 0 (1.1) E-mail addresses: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]. 2010 Mathematics Subject Classification. 11B73, 11M35, 26A48, 26D07, 34A05, 44A10.